We study efficient estimation of an interventional mean associated with a point exposure treatment under a causal graphical model represented by a directed acyclic graph without hidden variables. Under such a model, it may happen that a subset of the variables are uninformative in that failure to measure them neither precludes identification of the interventional mean nor changes the semiparametric variance bound for regular estimators of it. We develop a set of graphical criteria that are sound and complete for eliminating all the uninformative variables so that the cost of measuring them can be saved without sacrificing estimation efficiency, which could be useful when designing a planned observational or randomized study. Further, we construct a reduced directed acyclic graph on the set of informative variables only. We show that the interventional mean is identified from the marginal law by the g-formula (Robins, 1986) associated with the reduced graph, and the semiparametric variance bounds for estimating the interventional mean under the original and the reduced graphical model agree. This g-formula is an irreducible, efficient identifying formula in the sense that the nonparametric estimator of the formula, under regularity conditions, is asymptotically efficient under the original causal graphical model, and no formula with such property exists that only depends on a strict subset of the variables.
翻译:我们根据一个没有隐藏变量的定向循环图解代表的因果图形模型,研究与点暴露处理相关的干预平均值的有效估计。在这样一个模型下,可能发生这样的情况:一个子子变量是非信息化的,因为不能测量它们,既不能排除干预平均值的识别,也不能改变定期估测该变量的半参数差异。我们制定一套合理和完整的图形标准,以消除所有非信息化变量,从而在不牺牲估计效率的情况下可以节省测量这些变量的成本。在设计一项计划中的观测或随机化研究时,这种估算效率可能是有用的。此外,我们在一套信息化变量上只建立一个减少的定向循环图。我们表明,与减少的图形相关的g-公式(Robins,1986年)从边际法中确定了干预平均值,而用来估算原始和减少的图形模型所同意的干预平均值的半参数。这种公式是不可减损的,在设计一项规划的观测或随机化研究时可以有效地确定公式。此外,在常规条件下,我们从边际法中可以确定一个纯度的公式的公式,在原始的公式下,只能根据一个精确的公式的公式的公式,取决于一个有效的公式的公式。